Do I need a contract for my new job?

So I am going to leave my old job and got a letter from my new job that I can start either in a week or two weeks from this Monday. Do I need to get a contract from them so that I don't leave my old job and then get screwed if they decide that they change their mind? I am not scared but just don't know since my friend told me that there isn't any obligation to me if that happens.

Unless you have some sort of contract anything can happen

Please keep in mind that just because you ask for a contract does not mean they are obligated to give you one, and I'm not sure what I'd think about an employee who hasn't even started yet making that kind of request.

FWIW, I would not leave job 1 for job 2 without, at least, a written offer letter from job 2 which included a firm start date. Of course, if you are to be hired as an "at-will" employee, then job 2 could withdraw the offer notwithstanding the offer letter. Nonetheless, if a company issues an offer letter, it is usually committed to having the prospective employee start working on the start date included in said letter. In contrast, a verbal offer to start in a week or two appears rather casual at best or uncomfortably shaky at worst. Why leave the job you actually have on the basis of an amorphous promise for a job on an unspecified date?

Of course, you can "ask" for an employment contract. However, if your prospective employer planned to hire you on an at-will basis, virtually no chance exists it would then voluntarily consent to providing you with a written employment agreement. Indeed, as stated above, your prospective employer's representatives may consequently reconsider whether you are the "right fit" for them at all.